Abstract
We call attention to the fact that quantization includes the problem of a correct definition of quantum-mechanical observables like Hamiltonian, momentum, etc. as self-adjoint operators in an appropriate Hilbert space. This problem is nontrivial for systems on nontrivial manifolds or/and with singular interactions. A naïve treatment based on the experience in finite-dimensional algebra or even infinite-dimensional algebra with bounded operators can result in paradoxes and incorrect results. A set of such paradoxes is considered.
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Notes
- 1.
- 2.
For unbounded operators, there is a crucial difference between the notions of symmetric (Hermitian) and s.a. operators; for bounded operators, these notions actually coincide.
- 3.
S.a. according to Lagrange in mathematical terminology; see Chap. 4.
- 4.
Of course, a flat infinite space is also an idealization, as is any infinity.
- 5.
For a mathematician, quantization is a quantum deformation of classical structures; the deformation parameter is the Planck constant ℏ.
- 6.
The bar \(\overline {-}\) over an expression denotes complex conjugation.
- 7.
Numerous papers have been devoted to the study of various rules of assigning operators to classical quantities. A substantial contribution to a resolution of this problem is due to Berezin [25].
- 8.
It is rather a differential operation than a differential operator; see Chap. 4. A rigorous definition of the differentiation operator \(\hat{{d}}_{x}\) is given in the end of Sect. 2.3.4.
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Gitman, D.M., Tyutin, I.V., Voronov, B.L. (2012). Introduction. In: Self-adjoint Extensions in Quantum Mechanics. Progress in Mathematical Physics, vol 62. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4662-2_1
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DOI: https://doi.org/10.1007/978-0-8176-4662-2_1
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